Massachusetts Institute of Technology Physics 8 03 Fall 2004 Problem Set 7 Due Friday November 5 2004 at 4 PM Reading Assignment French pages 274 280 Beke Barrett see assignment 6 pages 189 250 new reading pages 208 251 Problem 7 1 Polarized radiation Write down the electric eld and associated magnetic eld in vacuum for travelling plane waves The amplitude of the electric vector is E0 and the frequency is a The radiation is linearly polarized in the y z plane at an angle of 45 with the y axis and it is travelling in the x direction There are two solutions b The radiation is circularly polarized in the y z plane and it is travelling in the x direction There are two solutions Problem 7 2 Linear polarizers Malus law absorption In lectures an envelope was handed out that contains three linear polarizers of type HN30 If 100 linearly polarized light of intensity I0 goes through an ide al polarizer properly aligned 100 linearly polarized light of intensity I0 will emerge Such a polarizer is called HN50 In reality polarizers are not ideal and they come in various types with di erent degrees of absor ption Your polarizers are of type HN30 This means that if you do the experiment as described above only 70 of I0 emerges Thus if we let the 100 linearly polarized light of intensity I0 go through two of th em both correctly aligned only 49 of I0 will emerge a Look through one of your linear polarizers at a light source Then place a second linear polarizer on top of the rst same orientation You will notice that the light intensity decreases Place two polarizers at right angle see the gure on the left Let us de ne the light intensity of the 100 linearly polarized light coming through the areas A B C and D as I0 The light intensity through area E is now zero Now stick your third linear polarizer between the two see the gure on the right and rotate this third polarizer in it s plane while keeping the original two at 90 degrees The darkness at E wi ll disappear Try it b What are the light intensities through the areas F G H K L M and N No tice the angle of 60 degrees Take into account that your polarizers are of type HN30 1 Problem 7 3 Radiation from an accelerated charge Do problem 4 1 from Bekefi and Barrett Electromagnetic Vibrations Waves and Radiation Cambridge MA The MIT Press September 15 1977 ISBN 0262520478 A charge q initially at rest is given a brief acceleration along the z axis Let this occur at the origin of the coordinate system at t 0 a Give the arrival times the relative strengths and specify the directions of the radiated electric eld seen by three observers in the y z plane at a large distance R from the origin One observer is on the y axis one on the z axis and one is on a radius making an angle of 30 to the z axis b Describe the orientation and magnitude of the associated radiated magnetic elds Problem 7 4 Radiation from an accelerated charge Do problem 4 2 from Bekefi and Barrett Electromagnetic Vibrations Waves and Radiation Cambridge MA The MIT Press September 15 1977 ISBN 0262520478 A point charge q has been moving with constant velocity w along a straight line until the time t t0 In the short time interval from t t0 to t t0 t a force perpendicular to the trajectory changes the direction without changing the magnitude of the velocity After the time t t0 t the charge again moves with velocity w along a straight line forming a small angle with the initial trajectory a What is the direction of the electric eld caused by the acceleration at the distant point P1 b In what direction is the radiation intensity of the accelerated charge most intense c Where is it least intense d Point P2 is twice as far from the bend in the trajectory as P1 By what fraction does the amplitude of the magnetic disturbance decrease as the radiation pulse moved from P1 to P2 e What is the total energy radiated Make careful sketches in answering parts a b c 2 Problem 7 5 Speed checked by radar A radar beam 3 cm is re ected o a car as shown in the gure head on re ection The radar transmitter is not moving The re ected beam is again received by the police the frequency f of this re ected signal is measured by a stationary receiver on the police car From this the velocity v of the car is calculated All this is done in a black box in less than a small fraction of a second A digital display tells the police o cer your speed nicely converted into miles hour a What is the frequency of this radar transmitter b Give an equation that relates f f and v Hint rst calculate the frequency that the car sees then send this signal back to the police Problem 7 6 Can t you hear the whistle blowing A train travels down a long straight track at a speed of 20 m sec 45 mph blowing its whistle constantly f 1000 Hz You the observer are standing at a point 100 m from the track see gure De ne time t 0 the moment that the train is closest to you a What frequency do you hear when i the train is far away and approaching ii the train is far away and receding and iii the train passes you at time t 0 b What frequency do you hear at time t 10 sec and at t 5 sec c Sketch the curve of the frequency that you hear versus time Problem 7 7 Our expanding universe simpli ed The gure below illustrates the famous expanding balloon analogy for our universe All of space is represented by the surface of the spherical balloon and clusters of galaxies are represented by spots painted on this surface the radius of the balloon corresponds to R t the radius of the universe As the balloon expands the spots remain at constant angular separations from one another Let the balloon expand at a constant rate a Prove that s t 3 1 R s R t 1 Here s is the separation between any two spots on the surface and s t is the speed of recession of one spot from another Note that eq 1 is equivalent to Hubble s Law v Hr 2 b What is H in terms of the symbols used in eq 1 Check that your answer is dimensionally correct H in general is expressed in km sec per Mpc The light photons from distant galaxies may be represented by ants crawling along the balloons surface at speed l c Show that for uniform cosmic expansion i e R t constant there is a distance s from beyond which these ants cannot ever reach our galaxy this distance is …